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58
qSchur algebras and complex reflection groups
"... Abstract. We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a qSchur algebra (parameter ̸ ∈ 1 2 + Z), providing thus character formulas for simple modules. We give some generalization to Bn(d). We prove an “abstract ” translation principle. These r ..."
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Cited by 58 (2 self)
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Abstract. We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a qSchur algebra (parameter ̸ ∈ 1 2 + Z), providing thus character formulas for simple modules. We give some generalization to Bn(d). We prove an “abstract ” translation principle. These results follow from the unicity of certain highest weight categories covering Hecke algebras. We also provide a semisimplicity criterion for Hecke algebras of complex reflection groups. 1.
Representations of rational Cherednik algebras
 CONTEMPORARY MATHEMATICS
"... This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category O. For type A, we explain relations with the Hilbert scheme of points on C². We insist on the analogy with ..."
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This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category O. For type A, we explain relations with the Hilbert scheme of points on C². We insist on the analogy with the representation theory of complex semisimple Lie algebras.
Cherednik algebras and Hilbert schemes in characteristic p
, 2006
"... We prove a localization theorem for the type An−1 rational Cherednik algebra Hc = H1,c(An−1) overFp, an algebraic closure of the finite field. In the most interesting special case where c ∈ Fp, we construct an Azumaya algebra Hc on Hilbn A2, the Hilbert scheme of n points in the plane, such that Γ(H ..."
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Cited by 39 (8 self)
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We prove a localization theorem for the type An−1 rational Cherednik algebra Hc = H1,c(An−1) overFp, an algebraic closure of the finite field. In the most interesting special case where c ∈ Fp, we construct an Azumaya algebra Hc on Hilbn A2, the Hilbert scheme of n points in the plane, such that Γ(Hilbn A2, Hc) = Hc. Our localization theorem provides an equivalence between the bounded derived categories of Hcmodules and sheaves of coherent Hcmodules on Hilbn A2, respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the HilbertChow morphism. This provides a link between our results and those of Bridgeland, King and Reid, and Haiman.
HARISHCHANDRA BIMODULES FOR QUANTIZED SLODOWY SLICES
"... Abstract. The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizations of the Poisson algebra of polynomial functions on the Slodowy slice. In this paper, we define and study HarishChandra bimodules ov ..."
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Cited by 34 (0 self)
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Abstract. The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizations of the Poisson algebra of polynomial functions on the Slodowy slice. In this paper, we define and study HarishChandra bimodules over Premet’s algebras. We apply the technique of HarishChandra bimodules to prove a conjecture of Premet concerning primitive ideals, to define projective functors, and to construct “noncommutative resolutions ” of Slodowy slices via translation functors. 1. Geometry of Slodowy slices 1.1. Introduction. Let
THE HILBERT SCHEME OF A PLANE CURVE SINGULARITY AND THE HOMFLY HOMOLOGY OF ITS LINK
, 2012
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Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras
, 2007
"... We relate the representations of the rational Cherednik algebras associated with the complex reflection group µℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Zalgebra construction. This is done so that as the parameters defining the Cherednik algebra va ..."
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Cited by 28 (2 self)
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We relate the representations of the rational Cherednik algebras associated with the complex reflection group µℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Zalgebra construction. This is done so that as the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change. This construction motivates us to use the geometry of the quiver varieties to interpret the ordering function (the cfunction) used to define a highest weight structure on category O of the Cherednik algebra. This interpretation provides a natural partial ordering on O which we expect will respect the highest weight structure. This partial ordering has appeared in a conjecture of Yvonne on the composition factors in O and so our results provide a small step towards a geometric picture for that. We also interpret geometrically another ordering function (the afunction) used in the study of Hecke algebras. (The connection between Cherednik algebras and Hecke algebras is provided by the KZfunctor.) This is related to a conjecture of Bonnafé and Geck on equivalence classes of weight functions for Hecke algebras with unequal parameters since the classes should (and do for type B) correspond to the G.I.T. chambers defining the quiver varieties. As a result anything that can be defined via the quiver varieties,
Drinfeld coproduct, quantum fusion tensor category and applications
, 2006
"... The class of quantum affinizations (or quantum loop algebras, see [Dr2, CP3, GKV, VV2, Mi1, N1, Jin, H3]) includes quantum affine algebras and quantum toroidal algebras. In general they have no Hopf algebra structure, but have a “coproduct ” (the Drinfeld coproduct) which does not produce tensor p ..."
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Cited by 22 (6 self)
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The class of quantum affinizations (or quantum loop algebras, see [Dr2, CP3, GKV, VV2, Mi1, N1, Jin, H3]) includes quantum affine algebras and quantum toroidal algebras. In general they have no Hopf algebra structure, but have a “coproduct ” (the Drinfeld coproduct) which does not produce tensor products of modules in the usual way because it is defined in a completion. In this paper we propose a new process to produce quantum fusion modules from it: for all quantum affinizations, we construct by deformation and renormalization a new (non semisimple) tensor category Mod. For quantum affine algebras this process is new and different from the usual tensor product. For general quantum affinizations, for example for toroidal algebras, so far, no process to produce fusion modules was known. We derive several applications from it: we construct the fusion of (finitely many) arbitrary lhighest weight modules, and prove that it is always cyclic. We establish exact sequences involving fusion of KirillovReshetikhin modules related to new Tsystems extending results of [N4, N3, H5]. Eventually for a large class of quantum affinizations we prove that the subcategory of finite length modules of Mod is stable
Quantizations of conical symplectic resolutions I: local and global structure
"... Abstract. We reexamine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical con ..."
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Cited by 19 (9 self)
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Abstract. We reexamine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a version of the BeilinsonBernstein localization theorem, a theory of HarishChandra bimodules and their relationship to convolution operators on cohomology, and a
HarishChandra homomorphisms and symplectic reflection algebras for wreathproducts
, 2005
"... The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreathproduct in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existen ..."
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Cited by 18 (2 self)
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The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreathproduct in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized
Generalized Jack polynomials and the representation theory of rational Cherednik algebras
 Selecta Math. (N.S
"... Abstract. We apply the DunklOpdam operators and generalized Jack polynomials to study category Oc for the rational Cherednik algebra of type G(r, p, n). We determine the set of aspherical values and, in case p = 1, answer a question of Iain Gordon on the ordering of category Oc. 1. ..."
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Cited by 17 (5 self)
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Abstract. We apply the DunklOpdam operators and generalized Jack polynomials to study category Oc for the rational Cherednik algebra of type G(r, p, n). We determine the set of aspherical values and, in case p = 1, answer a question of Iain Gordon on the ordering of category Oc. 1.